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Properties

Label 2-2040-17.13-c1-0-12
Degree $2$
Conductor $2040$
Sign $-0.346 - 0.938i$
Analytic cond. $16.2894$
Root an. cond. $4.03602$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−2.33 + 2.33i)7-s + 1.00i·9-s + (0.445 − 0.445i)11-s + 5.99·13-s + 1.00i·15-s + (−2.17 + 3.50i)17-s + 0.236i·19-s − 3.30·21-s + (3.45 − 3.45i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + (6.37 + 6.37i)29-s + (−3.61 − 3.61i)31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.316 + 0.316i)5-s + (−0.883 + 0.883i)7-s + 0.333i·9-s + (0.134 − 0.134i)11-s + 1.66·13-s + 0.258i·15-s + (−0.526 + 0.850i)17-s + 0.0543i·19-s − 0.721·21-s + (0.720 − 0.720i)23-s + 0.200i·25-s + (−0.136 + 0.136i)27-s + (1.18 + 1.18i)29-s + (−0.648 − 0.648i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.346 - 0.938i$
Analytic conductor: \(16.2894\)
Root analytic conductor: \(4.03602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :1/2),\ -0.346 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.850464398\)
\(L(\frac12)\) \(\approx\) \(1.850464398\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (2.17 - 3.50i)T \)
good7 \( 1 + (2.33 - 2.33i)T - 7iT^{2} \)
11 \( 1 + (-0.445 + 0.445i)T - 11iT^{2} \)
13 \( 1 - 5.99T + 13T^{2} \)
19 \( 1 - 0.236iT - 19T^{2} \)
23 \( 1 + (-3.45 + 3.45i)T - 23iT^{2} \)
29 \( 1 + (-6.37 - 6.37i)T + 29iT^{2} \)
31 \( 1 + (3.61 + 3.61i)T + 31iT^{2} \)
37 \( 1 + (1.31 + 1.31i)T + 37iT^{2} \)
41 \( 1 + (2.38 - 2.38i)T - 41iT^{2} \)
43 \( 1 - 6.86iT - 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 - 13.2iT - 53T^{2} \)
59 \( 1 + 11.7iT - 59T^{2} \)
61 \( 1 + (6.16 - 6.16i)T - 61iT^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + (-1.03 - 1.03i)T + 71iT^{2} \)
73 \( 1 + (3.16 + 3.16i)T + 73iT^{2} \)
79 \( 1 + (4.51 - 4.51i)T - 79iT^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-4.43 - 4.43i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.256265297007867898652825177331, −8.693055439228179026585817999848, −8.146287513169747033594176757294, −6.71961172018215923255008608664, −6.31306342298338448243307319868, −5.52386960348188572773243008555, −4.38178754540537986472156627000, −3.38576185917568058268939718814, −2.79952320441087899272839532230, −1.52008323387771449483111838653, 0.64345195332176021674983267754, 1.73852448052166247332373251151, 3.11262105114974685228413551009, 3.74081308049229583257174098624, 4.79146971727410976580793846802, 5.89791522854840580928401160812, 6.71126228263038018878175011469, 7.13221776022949087520641004260, 8.276790704939025616730146716972, 8.823156455684461089783255481812

Graph of the $Z$-function along the critical line